Linear spin-wave theory

Prerequisites

BdG & para-Hermitian diagonalization · DMI & exchange anisotropy

How to get from a classical magnetic ground state and a set of exchange tensors to a magnon dispersion ${\omega }_{\nu }(\mathbf{k})$, at the harmonic (linear) level. This is the back end of the pipeline I’m building: CRYSTAL23 hybrid+SOC → exchange tensors → LSWT.

1. The spin Hamiltonian

Start from a general bilinear spin model on a lattice,

$$\mathcal{H}=\frac{1}{2}\sum _{i\ne j}{\mathbf{S}}_i^{\top }{\mathbf{J}}_{ij}{\mathbf{S}}_j+\sum _i{\mathbf{S}}_i^{\top }{\mathbf{A}}_i{\mathbf{S}}_i,$$

where ${\mathbf{J}}_{ij}$ is a $3\times 3$ exchange tensor whose symmetric traceless part is the anisotropic (Kitaev/$\Gamma$) exchange, whose antisymmetric part encodes the Dzyaloshinskii–Moriya interaction ${\mathbf{D}}_{ij}$ via $\frac{1}{2}(J_{ij}^{\alpha \beta }-J_{ij}^{\beta \alpha })={ϵ}^{\alpha \beta \gamma }{D}_{ij}^{\gamma }$, and ${\mathbf{A}}_i$ is single-ion anisotropy. These are exactly the tensors a broken-symmetry hybrid+SOC calculation gives you — see exchange tensors from DFT.

2. Local frames and Holstein–Primakoff

Let the classical ground state assign each spin a direction ${\hat{\mathbf{n}}}_i$ (collinear or not). Rotate into a local frame $R_i$ such that the new ${\hat{z}}_i^{\prime }$ points along ${\hat{\mathbf{n}}}_i$, then apply the Holstein–Primakoff transformation to large $S$:

$$\begin{array}{rl}{S}_i^{z^{\prime }}& =S-a_i^{†}{a}_i,\\ S_i^{{+}^{\prime }}& =\sqrt{2S}(1-\frac{a_i^{†}{a}_i}{2S}{)}^{1/2}{a}_i\approx \sqrt{2S}{a}_i,\\ S_i^{{-}^{\prime }}& =\sqrt{2S}{a}_i^{†}(1-\frac{a_i^{†}{a}_i}{2S}{)}^{1/2}\approx \sqrt{2S}{a}_i^{†},\end{array}$$

with $[a_i,a_j^{†}]={\delta }_{ij}$. Linear spin-wave theory keeps terms up to quadratic in the bosons; the $1/S$ expansion parameter makes this controlled for large $S$. Substituting and keeping $\mathcal{O}(S)$ (classical energy), $\mathcal{O}(S^{1/2})$ (must vanish — it does iff ${\hat{\mathbf{n}}}_i$ is a stationary point), and $\mathcal{O}(S^0)$ (the magnons):

$$\mathcal{H}\approx E_{\text{cl}}+\frac{1}{2}\sum _{ij}[A_{ij}{a}_i^{†}{a}_j+A_{ij}^{\ast }{a}_i{a}_j^{†}+B_{ij}{a}_i^{†}{a}_j^{†}+B_{ij}^{\ast }{a}_i{a}_j].$$

The anomalous $B_{ij}$ terms are nonzero for antiferromagnets, frustrated, or non-collinear orders — that is what forces the Bogoliubov machinery below.

3. Fourier transform → BdG form

With $N$ magnetic sublattices, Fourier transforming gives a quadratic boson Hamiltonian in Bogoliubov–de Gennes (particle–hole doubled) form,

$$\mathcal{H}=E_0+\frac{1}{2}\sum _{\mathbf{k}}{\mathbf{\Psi }}_{\mathbf{k}}^{†}\mathcal{H}(\mathbf{k}){\mathbf{\Psi }}_{\mathbf{k}},{\mathbf{\Psi }}_{\mathbf{k}}=\left(\begin{array}{c}{a}_{1\mathbf{k}}\\ ⋮\\ a_{N\mathbf{k}}\\ a_{1,-\mathbf{k}}^{†}\\ ⋮\\ a_{N,-\mathbf{k}}^{†}\end{array}\right),\mathcal{H}(\mathbf{k})=\left(\begin{array}{cc}\mathbf{A}(\mathbf{k})& \mathbf{B}(\mathbf{k})\\ {\mathbf{B}}^{†}(\mathbf{k})& {\mathbf{A}}^{\top }(-\mathbf{k})\end{array}\right),$$

a $2N\times 2N$ para-Hermitian “grand dynamical matrix.”

4. Colpa diagonalization (do it right)

You cannot just diagonalize $\mathcal{H}(\mathbf{k})$ — the transformation must preserve bosonic commutation, i.e. be para-unitary with respect to $g=\mathrm{diag}(1_N,-1_N)$. The linear algebra is in BdG & para-Hermitian diagonalization; the recipe (Colpa’s method):

1. Check $\mathcal{H}(\mathbf{k})\succ 0$ (positive definite). If not, your assumed ground state is unstable — a real, useful diagnostic.
2. Cholesky factorize $\mathcal{H}(\mathbf{k})={\mathbf{K}}^{†}\mathbf{K}$.
3. Diagonalize the Hermitian matrix $\mathbf{K}g{\mathbf{K}}^{†}=\mathbf{U}\mathrm{diag}(\mathbf{\lambda }){\mathbf{U}}^{†}$.
4. The magnon energies are ${\omega }_{\nu }(\mathbf{k})=(g\mathbf{\lambda }{)}_{\nu }$ for the $N$ positive eigenvalues; the paraunitary $\mathbf{T}$ satisfying ${\mathbf{T}}^{†}\mathcal{H}\mathbf{T}=\mathrm{diag}(\mathbf{\omega },\mathbf{\omega })$ and ${\mathbf{T}}^{†}g\mathbf{T}=g$ is $\mathbf{T}={\mathbf{K}}^{-1}\mathbf{U}(g\mathrm{diag}(\mathbf{\lambda }){)}^{1/2}$.

5. Why this matters for DFT

For a Heisenberg ferromagnet this all collapses to the familiar $\omega (\mathbf{k})=\frac{2}{\hslash }[J(0)-J(\mathbf{k})]S$. The point of the full tensorial LSWT is that anisotropy and DMI enter the dispersion, gap the Goldstone mode (and sidestep Mermin–Wagner in 2D), and are precisely the quantities semilocal DFT and DFT+U get wrong. Feeding hybrid+SOC exchange tensors into this engine is the whole bet.

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See also: Physics index · exchange tensors from broken-symmetry DFT

Key references

• Toth & Lake, J. Phys.: Condens. Matter 27, 166002 (2015) — SpinW.
• Colpa, Physica A 93, 327 (1978) — paraunitary diagonalization.